Finite Group Databasebeta

← #6526684

Subgroups — #6526684

Order 512.

2424 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3226C211yesmaximal, characteristic, complemented
2C2×#3100C211yesmaximal, characteristic
3C2×#2542C211yesmaximal, characteristic, complemented
4#34206C211yesmaximal, characteristic, complemented
5#28114C214yesmaximal, complemented
6#27644C214yesmaximal, complemented
7#25449C214yesmaximal, complemented
8#25318C214yesmaximal, complemented
9#24811C214yesmaximal, complemented
10#24676C214yesmaximal, complemented
11#21951C211yesmaximal, characteristic, complemented
12#21833C211yesmaximal, characteristic, complemented
13#20732C211yesmaximal, characteristic, complemented
14C2×#532C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C22⋊Q8—21no—
17C22×C22⋊Q8—21no—
18C2×#520—21no—
19C22×C4×D4C2211yescharacteristic
20#3226—28no—
21C2×#500C2212yescomplemented
22C2×#486C2212yescomplemented
23#3100—28no—
24C2×#480C2212yes—
25C2×#480C2214yes—
26C2×#474C2212yes—
27C2×#474C2214yescomplemented
28C2×#459C2212yes—
29C2×#447C2212yes—
30C2×C4×D8C2214yes—
31C2×#431C2211yescharacteristic
32C2×#431C2211yescharacteristic
33#2542—22no—
34#2542—22no—
35#2542C2214yes—
36#2542C2214yes—
37C2×#399—22no—
38#2367—22no—
39#2366C2212yescomplemented
40#2365C2212yescomplemented
41#2354C2212yescomplemented
42#2352—22no—
43D4×C4⋊C4C2212yescomplemented
44#2325—22no—
45#2311—21no—
46C4×#121C2214yes—
47C2×#390C2212yes—
48#2285C2211yescharacteristic
49C2×#384C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2083C2218yescomplemented
52#2071C2218yescomplemented
53#2067C2218yes—
54#2054C2218yes—
55#1990C2212yescomplemented
56#1990C2212yescomplemented
57#1988C2214yescomplemented
58#1947C2218yescomplemented
59#1939C2214yescomplemented
60#1939C2214yescomplemented
61#1925C2218yes—
62#1879C2218yes—
63#1823C2214yes—
64#1823C2212yes—
65#1802C2212yes—
66#1802C2212yes—
67#1802C2218yescomplemented
68#1797C2214yes—
69#1786C2212yes—
70C4×#102C2214yes—
71C22×#41—21no—
72C23×Q8—21no—
73C23×D4C2311yescharacteristic
74C24×C4—21no—
75C22×D8C2314yes—
76#532—24no—
77#532D418yes—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×#127C2311yescharacteristic
86C2×#124—21no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—48no—
92C2×C22⋊Q8—48no—
93C2×#121—21no—
94C2×#121C2312yes—
95#520—48no—
96C2×C4×Q8—22no—
97C2×C4×D4—24no—
98C2×C4×D4—22no—
99C2×C4×D4—21no—
100C2×C4×D4—21no—
101C2×C4×D4—21no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4C2312yes—
105C2×C4×D4C2312yes—
106C2×C4×D4C2312yes—
107C2×C4×D4C2314yes—
108C2×#117—21no—
109C2×#117—22no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—22no—
120C22×C22⋊C4—22no—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123#500—28no—
124#486—28no—
125#480—28no—
126#480—216no—
127#474—28no—
128#474—216no—
129#459—28no—
130#447—28no—
131C4×D8—216no—
132#431—24no—
133#431—24no—
134C2×#107C2318yes—
135C2×#107C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2318yes—
139C2×#102C2314yes—
140C2×#102C2318yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2312yes—
148#399—28no—
149#399—28no—
150#391—22no—
151#390—22no—
152#390—24no—
153#390C2314yes—
154#390C2318yes—
155#389C2314yes—
156#388—24no—
157#388—24no—
158#387—24no—
159#386—24no—
160#386—24no—
161#385—24no—
162#384—24no—
163#384C2311yescharacteristic
164#384C2311yescharacteristic
165#384C2318yes—
166#382—21no—
167#380—21no—
168#380—22no—
169#380—22no—
170#380C2312yes—
171C4×C4⋊C4—21no—
172C4×C4⋊C4D412yes—
173C4×C4⋊C4D412yes—
174C4×C4⋊C4C2314yes—
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2312yes—
178C4×C22⋊C4—24no—
179C2×#95D412yes—
180#340C2318yes—
181#340C2314yes—
182#340C2314yes—
183#340C2318yes—
184#336C2318yes—
185C25—21no—
186C2×#41—28no—
187C2×#41—44no—
188C2×#41—44no—
189C22×Q8—21no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—44no—
193C22×Q8—44no—
194C22×Q8—44no—
195C22×D4—22no—
196C22×D4—24no—
197C22×D4—21no—
198C22×D4—21no—
199C22×D4—21no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—42no—
203C22×D4C2412yes—
204C22×D4C2412yes—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—21no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4C2×D412yes—
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D412yes—
217C23×C4C2×D411yescharacteristic
218C23×C4C2411yescharacteristic
219C2×D8—28no—
220C2×D8—28no—
221C2×D8—44no—
222C22×C8C2414yes—
223C22×C8C2414yes—
224C4⋊Q8—24no—
225C4⋊Q8—24no—
226C4⋊Q8—22no—
227C4⋊Q8—24no—
228C4⋊Q8—24no—
229C4⋊Q8—44no—
230C4⋊Q8—44no—
231C4⋊Q8C2×D414yes—
232#127—22no—
233#127C2×D414yes—
234#124—28no—
235C22⋊Q8—416no—
236C22⋊Q8—416no—
237C22⋊Q8—416no—
238C22⋊Q8—416no—
239#121—28no—
240#121—24no—
241#121C2×D418yes—
242C4×Q8—48no—
243C4×D4—28no—
244C4×D4—24no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—24no—
249C4×D4—48no—
250C4×D4—416no—
251C4×D4—48no—
252C4×D4—44no—
253C4×D4—44no—
254C4×D4C2×D418yes—
255#117—44no—
256#117—48no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—48no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—48no—
277C2×C4⋊C4—48no—
278C2×C4⋊C4D818yescomplemented
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2414yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—48no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—48no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—44no—
304C2×C42C2×D414yes—
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D414yes—
308C2×C42#4114yes—
309C2×C42C2414yes—
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312#107—28no—
313#107—216no—
314C4⋊C8—24no—
315C4⋊C8—24no—
316C4⋊C8—48no—
317#102—28no—
318#102—216no—
319#102—28no—
320#102—28no—
321#102—44no—
322C22⋊C8—24no—
323C22⋊C8—24no—
324C4×C8—24no—
325C4×C8—24no—
326#95—22no—
327#95—24no—
328#95—22no—
329#95—24no—
330#95C2×D414yes—
331#95C2×D414yes—
332C24—24no—
333C24—22no—
334C24—22no—
335C24—42no—
336C24—42no—
337C24—42no—
338C24—42no—
339C24—42no—
340C24—42no—
341C24C22×D411yescharacteristic
342#41—432no—
343C2×Q8—24no—
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8—42no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—42no—
350C2×Q8—44no—
351C2×Q8—44no—
352C2×Q8—816no—
353C2×D4—24no—
354C2×D4—28no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—24no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—42no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—81no—
374C2×D4—81no—
375C2×D4C22×D414yes—
376C2×D42−1+414yes—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—28no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—48no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4—816no—
410C22×C4#12114yes—
411C22×C4#12114yes—
412C22×C4C2×D818yes—
413C22×C4C2×D814yes—
414C22×C4AGL(1,ℤ/8)14yes—
415C22×C4C22×D414yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D412yes—
420C22×C4C22×D414yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C2×#4112yes—
424C22×C4C2511yescharacteristic
425D8—416no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—416no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—42no—
448C4⋊C4—48no—
449C4⋊C4—416no—
450C4⋊C4—42no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4C2×D818yes—
454C4⋊C4C2×D818yes—
455C4⋊C4C22×D414yes—
456C22⋊C4—28no—
457C22⋊C4—28no—
458C22⋊C4—416no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—416no—
463C22⋊C4—48no—
464C22⋊C4—84no—
465C22⋊C4—84no—
466C42—28no—
467C42—44no—
468C42—48no—
469C42—44no—
470C42#12114yes—
471C42#12114yes—
472C42C22×D412yes—
473C42C22×D412yes—
474C42C2×#4112yes—
475C42C2×#4112yes—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—24no—
481C23—24no—
482C23—48no—
483C23—48no—
484C23—48no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—84no—
498C23—84no—
499C23—84no—
500C23—84no—
501C23C2×#12112yes—
502C23C22×D812yes—
503C23C2×AGL(1,ℤ/8)12yes—
504C23C23×D411yescharacteristic
505Q8—48no—
506Q8—88no—
507Q8—88no—
508D4—24no—
509D4—28no—
510D4—24no—
511D4—48no—
512D4—416no—
513D4—42no—
514D4—44no—
515D4—42no—
516D4—48no—
517D4—84no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—24no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—44no—
533C2×C4—48no—
534C2×C4—48no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—416no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—84no—
551C2×C4—88no—
552C2×C4—88no—
553C2×C4—88no—
554C2×C4—168no—
555C2×C4#459116yes—
556C2×C4C2×#12112yes—
557C2×C4C2×#12112yes—
558C2×C4C2×#12112yes—
559C2×C4C2×#12112yes—
560C2×C4C2×#12112yes—
561C2×C4C22×D818yes—
562C2×C4C22×D812yes—
563C2×C4C22×D812yes—
564C2×C4C2×AGL(1,ℤ/8)12yes—
565C2×C4C2×AGL(1,ℤ/8)12yes—
566C2×C4C23×D411yescharacteristic
567C2×C4C22×#4111yescharacteristic
568C2×C4C2×2−1+412yes—
569C2×C4#58512yes—
570C8—44no—
571C8—44no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—816no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22C2×#45914yes—
594C22C22×#12111yescharacteristic
595C22#348511yescharacteristic
596C22#348611yescharacteristic
597C22C23×D811yescharacteristic
598C22C22×AGL(1,ℤ/8)11yescharacteristic
599C22#358611yescharacteristic
600C22#358611yescharacteristic
601C4—22no—
602C4—22no—
603C4—22no—
604C4—28no—
605C4—44no—
606C4—44no—
607C4—82no—
608C4—82no—
609C4—84no—
610C4—82no—
611C4C2×#45914yes—
612C4C2×#45914yes—
613C4#348912yes—
614C4#349512yes—
615C2—24no—
616C2—84no—
617C2—84no—
618C2#3563311yescharacteristic
619C2C22×#45911yescharacteristic
620C2#6042711yescharacteristic
621C2#6169411yescharacteristic
622C2#6171711yescharacteristic
623C2#6181711yescharacteristic
624C2#6182711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3100C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#3226C2semidirect
3. C2×#2542C2semidirect
4. #34206C2semidirect
5. #28114C2semidirect
6. #27644C2semidirect
7. #25449C2semidirect
8. #25318C2semidirect
9. #24811C2semidirect
10. #24676C2semidirect
11. #21951C2semidirect
12. #21833C2semidirect
13. #20732C2semidirect
21. C2×#500C22semidirect
22. C2×#486C22semidirect
27. C2×#474C22semidirect
39. #2366C22semidirect
40. #2365C22semidirect
41. #2354C22semidirect
43. D4×C4⋊C4C22semidirect
51. #2083C22semidirect
52. #2071C22semidirect
55. #1990C22semidirect
56. #1990C22semidirect
57. #1988C22semidirect
58. #1947C22semidirect
59. #1939C22semidirect
60. #1939C22semidirect
67. #1802C22semidirect
278. C2×C4⋊C4D8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#32262
C2×#31002
C2×#25422
#342062
#281142
#276442
#254492
#253182
#248112
#246762
#219512
#218332
#207322